integral ((x-2)^3)/(3x^2)

Lösung

integral

\frac{( x - 2 ) ^{3}}{3 x ^{2}}

\frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x}) + C

Schritte zur Lösung

\int \frac{( x - 2 ) ^{3}}{3 x ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{( x - 2 ) ^{3}}{x ^{2}} d x

Multipliziere aus

\frac{( x - 2 ) ^{3}}{x ^{2}} : x - 6 + \frac{12}{x} - \frac{8}{x ^{2}}

= \frac{1}{3} \cdot \int x - 6 + \frac{12}{x} - \frac{8}{x ^{2}} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int x d x - \int 6 d x + \int \frac{12}{x} d x - \int \frac{8}{x ^{2}} d x)

\int x d x = \frac{x ^{2}}{2}

\int 6 d x = 6 x

\int \frac{12}{x} d x = 12 ln ∣ x ∣

\int \frac{8}{x ^{2}} d x = - \frac{8}{x}

= \frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ - (- \frac{8}{x}))

Vereinfache

= \frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (\frac{x ^{2}}{2} - 6 x + 12 ln ∣ x ∣ + \frac{8}{x}) + C

in t e g r a l \frac{e ^{3} x}{3}

in t e g r a l 7 x^{2}

in t e g r a l y = 12 (x - 1)^{2} (x + 2)

in t e g r a l lo g_{10} (x^{2} + 1)

in t e g r a l (1 + x^{2})^{0.5}